Faster transforms using early aborts and precision refinements

ABSTRACT

Fast transforms that use early aborts and precision refinements are disclosed. When to perform a corrective action is detected based upon testing the incremental calculations of transform coefficients. Corrective action is then performed. The corrective action includes refining the incremental calculations to obtain additional precision and/or aborting the incremental calculations when the resulting numbers are sufficient.

RELATED PATENT DOCUMENTS

This is a divisional of patent application Ser. No. 09/694,455, filed onOct. 23, 2000 (BLD920000064US1), to which Applicant claims priorityunder 35 U.S.C. §120. This application is also related to the followingco-pending and commonly-assigned patent applications, which are herebyincorporated herein by reference in their respective entirety:

“FASTER DISCRETE COSINE TRANSFORMS USING SCALED TERMS” to Brady et al.,having patent application Ser. No. 09/694,452, filed on Oct. 23, 2000(attorney docket no. BLD9-2000-0056US1).

“FASTER TRANSFORMS USING SCALED TERMS” to Trelewicz et al., havingpatent application Ser. No. 09/694,448, filed on Oct. 23, 2000 (attorneydocket no. BLD9-2000-0059US1).

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates in general to data processing, and moreparticularly to faster transforms that use early aborts and precisionrefinements.

2. Description of Related Art

Transforms, which take data from one domain (e.g., sampled data) toanother (e.g., frequency space), are used in many signal and/or imageprocessing applications. Such transforms are used for a variety ofapplications, including, but not limited to data analysis, featureidentification and/or extraction, signal correlation, data compression,or data embedding. Many of these transforms require efficientimplementation for real-time and/or fast execution where compression mayor may not be used.

Data compression is desirable in many data handling processes, where toomuch data is present for practical applications using the data.Commonly, compression is used in communication links, to reducetransmission time or required bandwidth. Similarly, compression ispreferred in image storage systems, including digital printers andcopiers, where “pages” of a document to be printed may be storedtemporarily in memory. Here the amount of media space on which the imagedata is stored can be substantially reduced with compression. Generallyspeaking, scanned images, i.e., electronic representations of hard copydocuments, are often large, and thus make desirable candidates forcompression.

In data processing, data is typically represented as a sampled discretefunction. The discrete representation is either made deterministicallyor statistically. In a deterministic representation, the pointproperties of the data are considered, whereas, in a statisticalrepresentation, the average properties of the data are specified. Inparticular examples referred to herein, the terms images and imageprocessing will be used. However, those skilled in the art willrecognize that the present invention is not meant to be limited toprocessing images but is applicable to processing different data, suchas audio data, scientific data, image data, etc.

In a digital image processing system, digital image signals are formedby first dividing a two-dimensional image into a grid. Each pictureelement, or pixel, in the grid has associated therewith a number ofvisual characteristics, such as brightness and color. Thesecharacteristics are converted into numeric form. The digital imagesignal is then formed by assembling the numbers associated with eachpixel in the image into a sequence which can be interpreted by areceiver of the digital image signal.

Signal and image processing frequently require converting the input datainto transform coefficients for the purposes of analysis. Often only aquantized version of the coefficients is needed (e.g. JPEG/MPEG datacompression or audio/voice compression). Many such applications need tobe done fast in real time such as the generation of JPEG data for highspeed printers.

Pressure is on the data signal processing industry to find the fastestmethod by which to most effectively and quickly perform the digitalsignal processing. As in the field of compression generally, research ishighly active and competitive in the field of fast transformimplementation. Researchers have made a wide variety of attempts toexploit the strengths of the hardware intended to implement thetransforms by exploiting properties found in the transform and inversetransform.

One such technique is the ISO 10918-1 JPEG International Standard/ITU-TRecommendation T.81. The draft JPEG standard is reproduced in Pennebakerand Mitchell, JPEG: Still Image Data Compression Standard, New York, VanNostrand Reinhold, 1993, incorporated herein by reference. Onecompression method defined in the JPEG standard, as well as otheremerging compression standards, is discrete cosine transform (DCT)coding. Images compressed using DCT coding are decompressed using aninverse transform known as the inverse DCT (IDCT). An excellent generalreference on DCTs is Rao and Yip, Discrete Cosine Transform, New York,Academic Press, 1990, incorporated herein by reference. It will beassumed that those of ordinary skill in this art are familiar with thecontents of the above-referenced books.

It is readily apparent that if still images present storage problems forcomputer users and others, motion picture storage problems are far moresevere, because full-motion video may require up to 60 images for eachsecond of displayed motion pictures. Therefore, motion picturecompression techniques have been the subject of yet further developmentand standardization activity. Two important standards are ISO 11172 MPEGInternational Standard and ITU-T Recommendation H.261. Both of thesestandards rely in part on DCT coding and IDCT decoding.

However, research generally focuses on specific techniques, such as theabove-mentioned techniques that used DCT coding to provide the desireddegree of compression. Nevertheless, other transforms may be used toprovide certain advantages under certain circumstances. For example, inthe DCT compression coding method discussed above, an input image isdivided into many uniform blocks and the two-dimensional cosinetransform function is applied to each block to transform the datasamples into a set of transform coefficients to remove the spatialredundancy. However, even though a high compression rate may beattained, a blocking effect, which may be subtle or obvious, isgenerated. Further, vector quantization methods that may be utilized bythe compression system are advantageous due to their contribution to thehigh compression rate. On the other hand, a sub-band method may reducethe blocking effect which occurs during high rates of data compression.The wavelet transform (WT) or Sub-Band Coding (SBC) methods encodesignals based on, for example, time and frequency components. As such,these transform methods can be useful for analyzing non-stationarysignals and have the advantage that they may be designed to take intoaccount the characteristics of the human visual system (HVS) for imageanalysis.

Scaled terms may be used to replace multiplicative constants like cosineterms in a Discrete Cosine Transform (DCT) with a minimum number ofadditions/subtractions. However, the scaled terms merely approximate theconstants in the transform equations. Thus, some error is accepted tokeep the precision confined to a fixed number of bits or to minimize thenumber of operations. If the resulting numbers are further from adecision boundary (e.g., a threshold value or a quantization boundary)than the maximum possible error, the result will not be affected by theapproximations. However, the resulting numbers may be determined, duringthe incremental calculations, to require additional precision. Yet, theoriginal input values are no longer available in the registers, andrefetching the original input values from memory can impose cyclesassociated with cache misses and memory latency. The brute-force optionis to perform an inverse transform (e.g., an IDCT) on the values, andthen re-run the forward transform (e.g., FDCT, sometimes denoted justDCT) with higher precision. The disadvantage of the brute force approachis that operations are wasted.

It can be seen then that there is a need to provide faster transformsthat use early aborts and precision refinements to save processingcycles thereby providing faster transform calculations and decreasedexecution times.

SUMMARY OF THE INVENTION

To overcome the limitations in the prior art described above, and toovercome other limitations thatc will become apparent upon reading andunderstanding the present specification, the present invention disclosesfaster transforms that use early aborts and precision refinements.

The present invention solves the above-described problems by detectingwhen to perform a corrective action based upon testing the incrementalcalculations of transform constants and performing the correctiveaction: refining the incremental calculations to obtain additionalprecision and/or aborting the incremental calculations when theresulting number is going to be too small. Those skilled in the art willrecognize that throughout this specification, the term “matrix” is usedin both its traditional mathematical sense and also to cover allhardware and software systems which when analyzed could be equivalentlyrepresented as a mathematical matrix.

A method in accordance with the principles of the present inventionincludes testing at least one number resulting from an incrementalcalculation of transform coefficients during a transform, determiningwhether to perform a corrective action based upon the testing andperforming the corrective action when a corrective action is determinedto be needed.

Other embodiments of a method in accordance with the principles of theinvention may include alternative or optional additional aspects. Onesuch aspect of the present invention is that the determining comprisesdetecting whether the incremental calculation of the transformcoefficients will result in transform coefficients with unacceptableprecision and the performing corrective action comprises refining the atleast one number.

Another aspect of the present invention is that the transform comprisesa transform matrix and wherein the refining comprises applying arefinement matrix for increasing precision of the incrementalcalculation of the transform constants.

Another aspect of the present invention is that the refinement matrixcomprises I+_(d)D_(m+1)D_(m) ⁻¹.

Another aspect of the present invention is that the method furtherincludes generating at least one refinement matrix based onapproximately calculated transform constants.

Another aspect of the present invention is that the generating at leastone refinement matrix is performed offline or at initialization.

Another aspect of the present invention is that the generating therefinement matrix comprises recognizing that an approximate transform isinvertible, generating the refinement matrix given by I+_(d)D_(m+1)D_(m)⁻¹, and structuring the transform for efficient computation.

Another aspect of the present invention is that the generating therefinement matrix includes recognizing that recovery of the nth columnof a transform matrix for generating the transform is impossible,calculating a pseudo inverse for a portion of the transform matrix andgenerating an approximation for the refinement matrix using the pseudoinverse for the transform matrix.

Another aspect of the present invention is that the approximation of therefinement matrix comprises I+_(d)D_(1d){tilde over (D)}₀.

Another aspect of the present invention is that the determining furthercomprises determining whether an error resulting from terminating theincremental calculation is acceptable and the performing correctiveaction comprises aborting the incremental calculation of a transformcoefficient.

Another aspect of the present invention is that the incrementalcalculation is terminated when a determination is made that theincremental calculation will result in a number that is projected to bewithin a predetermined range.

Another aspect of the present invention is that the number that isprojected to be within a predetermined range comprises a transformcoefficient that does satisfy a precision requirement.

Another aspect of the present invention is that the incrementalcalculation is terminated when a refinement to the transform coefficientis determined not to change the result.

Another aspect of the present invention is that a refinement to thetransform coefficient is determined not to change the result when, afterchecking the relative magnitudes of the results of the incrementalcalculations, an intermediate calculation of at least one transformcoefficient is small compared to the intermediate calculation of anothertransform coefficient.

Another aspect of the present invention is that a refinement to thetransform coefficient is determined not to change the result when, afterchecking the magnitude of the results of at least one incrementalcalculation, at least one intermediate calculation of the transformcoefficient is less than a predetermined threshold.

Another aspect of the present invention is that the determining furthercomprises determining that a transform coefficient is going to be withina predetermined range of zero and the performing corrective actioncomprises aborting the incremental calculation of the transformcoefficient.

In another embodiment of the present invention, a data compressionsystem is provided. The data compression system includes a transformerfor applying a linear analysis transform to decorrelate data intotransform coefficients using transform equations, the transformerreducing errors of the transform by testing at least one numberresulting from an incremental calculation of transform coefficientsduring a transform, determining whether to perform a corrective actionbased upon the testing and performing the corrective action when acorrective action is determined to be needed.

In another embodiment of the present invention, a printer is provided.The printer includes memory for storing image data, a processor forprocessing the image data to provide a print stream output and aprinthead driving circuit for controlling a printhead to generate aprintout of the image data, wherein the processor reduces errors of thetransform by testing at least one number resulting from an incrementalcalculation of transform coefficients during a transform, determiningwhether to perform a corrective action based upon the testing andperforming the corrective action when a corrective action is determinedto be needed.

In another embodiment of the present invention, an article ofmanufacture is provided. The article of manufacture includes a programstorage medium readable by a computer, the medium tangibly embodying oneor more programs of instructions executable by the computer to perform amethod for reducing errors during data processing, the method includingtesting at least one number resulting from an incremental calculation oftransform coefficients during a transform, determining whether toperform a corrective action based upon the testing and performing thecorrective action when a corrective action is determined to be needed.

In another embodiment of the present invention, a data analysis systemis provided. The data analysis system includes transform equationsformed by testing at least one number resulting from an incrementalcalculation of transform coefficients during a transform, determiningwhether to perform a corrective action based upon the testing andperforming the corrective action when a corrective action is determinedto be needed and a transformer for applying the transform equations toperform a linear transform to decorrelate data into transformcoefficients.

These and various other advantages and features of novelty whichcharacterize the invention are pointed out with particularity in theclaims annexed hereto and form a part hereof. However, for a betterunderstanding of the invention, its advantages, and the objects obtainedby its use, reference should be made to the drawings which form afurther part hereof, and to accompanying descriptive matter, in whichthere are illustrated and described specific examples of an apparatus inaccordance with the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

Referring now to the drawings in which like reference numbers representcorresponding parts throughout:

FIG. 1 illustrates a typical image compression system;

FIG. 2 illustrates a flow chart of a method for providing fastertransforms using scaled terms;

FIG. 3 illustrates a flow chart for providing faster transforms usingcorrective action to provide faster transform calculations and decreasedexecution times;

FIG. 4 illustrates a flow chart of the abort method according to thepresent invention that demonstrates aborting further iterations of thetransform coefficient calculation process;

FIG. 5 is illustrates the testing of the at least one incrementallycalculated number;

FIG. 6 is a flow chart of the refinement method according to the presentinvention;

FIG. 7 illustrates a flow chart of a first method for generating arefinement matrix;

FIG. 8 is a flow chart showing a second method for generating arefinement matrix when _(d)D₀ is not invertible;

FIG. 9 illustrates a printer according to the present invention;

FIG. 10 illustrates a data analyzing system according to the presentinvention; and

FIG. 11 illustrates another data analyzing system according to thepresent invention.

DETAILED DESCRIPTION OF THE INVENTION

In the following description of the exemplary embodiment, reference ismade to the accompanying drawings which form a part hereof, and in whichis shown by way of illustration the specific embodiment in which theinvention may be practiced. It is to be understood that otherembodiments may be utilized as structural changes may be made withoutdeparting from the scope of the present invention.

The present invention provides faster transforms that use early abortsand precision refinements. Faster transforms are obtained by detectingwhen to perform a corrective action based upon testing the incrementalcalculations of transform coefficients and performing the correctiveaction: refining the incremental calculations to obtain additionalprecision and/or aborting the incremental calculations when at least oneresulting number is going to be too small.

FIG. 1 illustrates a typical image compression system 100. The datacompression system may include three closely connected components namely(a) Transformer 120, (b) Quantizer 130, and (c) Optional Entropy Encoder140. Compression is accomplished by applying a linear transform todecorrelate the image data 110, quantizing the resulting transformcoefficients, and, if desired, entropy coding the quantized values. Avariety of linear transforms have been developed which include DiscreteFourier Transform (DFT), Discrete Cosine Transform (DCT), DiscreteWavelet Transform (DWT) and many more, each with its own advantages anddisadvantages.

The quantizer 130 simply reduces the number of bits needed to store thetransformed coefficients by reducing the precision of those values.Since this is a many-to-one mapping, it is a lossy process and is thesignificant source of compression in such an encoder. Quantization canbe performed on each individual coefficient, which is known as ScalarQuantization (SQ). Quantization can also be performed on a collection ofcoefficients together, and this is known as Vector Quantization (VQ).Both uniform and non-uniform quantizers can be used depending on theproblem at hand.

The optional entropy encoder 140 further compresses the quantized valueslosslessly to give better overall compression. It may use a model toaccurately determine the probabilities for each quantized value andproduces an appropriate code based on these probabilities so that theresultant output code stream will be smaller than the input stream. Themost commonly used entropy encoders are the Huffman encoder and thearithmetic encoder, although for applications requiring fast execution,simple run-length encoding (RLE) has proven very effective.

The term image transforms usually refers to a class of unitary matricesused for representing images. This means that images can be converted toan alternate representation using these matrices. These transforms formthe basis of transform coding. Transform coding is a process in whichthe coefficients from a transform are coded for transmission.

The signal F(x) is a function mapping each integer from 0 . . . n−1 intoa complex number. An example is given by a line of a sampled orpixelated image, where the samples or pixels are equally spaced. An“orthogonal basis” for a collection of such F(x) is a set{b_(y)(x)}_(y=0) ^(n−1) of functions, where

${\sum\limits_{x = 0}^{n - 1}{{b_{y}(x)}{b_{z}(x)}}} = {{0\mspace{14mu} {for}\mspace{14mu} y} \neq}$

z. A “transform” of F(x), denoted {circumflex over (F)}(y), is given by

${\hat{F}(y)} = {\sum\limits_{x = 0}^{n - 1}{{F(x)}{{b_{y}(x)}.}}}$

Transforms of this type are used in many signal and image processingapplications to extract information from the original signal F. Oneexample of a transform is the discrete Fourier transform (DFT), whereb_(y)(x)=exp(2πixy/n). A related example is the discrete cosinetransform (DCT), where b_(y)(x)=cos(2πxy/n). Another example is thewavelet transform, where b_(y)(x) is a particular scaled and offsetversion of the mother wavelet function. (See, Ingrid Daubechies, TenLectures on Wavelets, Society for Industrial & Applied Mathematics, (May1992)).

The theoretical basis for the independent scaling operations will now bedemonstrated by showing the mathematical basis for being able to performthe scales without destroying the structure of the transform. Define atransform

${\hat{F}(y)} = {\sum\limits_{x = 0}^{n - 1}{{F(x)}{{b_{y}(x)}.}}}$

Consider those cases (described below) when the b_(y)(x) are such thatthis transform can be split into two or more disjoint sums, regardlessof the structure of F(x). (The term “disjoint”, when used herein inreference to the sets of equations, means that there are no transformcoefficients in common between equations in the two disjoint sets ofequations.) For example, if b_(2y)(x) have even symmetry, andb_(2y+1)(x) have odd symmetry, it is known from mathematics that anyF(x) can be written uniquely as F(x)=F_(e)(X)+F_(o)(x), where F_(e)(x)is even (symmetric about zero) and F_(o)(x) is odd (anti-symmetric aboutzero), and that

${\sum\limits_{x}^{\;}{{F_{e}(x)}{b_{{2\; y} - 1}(x)}}} = {{\sum\limits_{x}^{\;}{{F_{o}(x)}{b_{2\; y}(x)}}} = 0.}$

This enables the transform to be written equivalently as:

${\hat{F}(y)} = {{\sum\limits_{y = 0}^{\lfloor{{({n - 1})}/2}\rfloor}{{F_{e}(x)}{b_{2\; y}(x)}}} + {\sum\limits_{y = 1}^{\lfloor{n/2}\rfloor}{{F_{o}(x)}{b_{{2\; y} - 1}(x)}}}}$

FIG. 2 illustrates a flow chart 200 of a method for providing fastertransforms using scaled terms. In FIG. 2, transform equations are splitinto at least one sub-transform having at least two transform constants210. The term “sub-transforms”, as used herein, references thecollection of equations used to generate a subset of the transformedterms, where the subset may contain all of the transformed terms, orfewer that the total number of transformed terms. Next, the transformconstants for each collection are scaled independently of the othercollections with a scaling term to maintain a substantially uniformratio between the transform constants within the collection, wherein thescaling term may be chosen according to a predetermined cost function220. The result is the transform equations for transforming the block.Then, data is separated into at least one block 230. The block is thentransformed into transformed data using the transform equations 240.Referring to the quantizer 130 of FIG. 1, the transformed data may thenbe quantized by incorporating the scaling into the quantization.Choosing the scaled term for the constants requires the use of a costfunction that represents the needs of the target system.

Scaled terms may be used to replace multiplicative constants like cosineterms in a Discrete Cosine Transform (DCT) with a minimum number ofadditions/subtractions. For a 1-D DCT, an 8×1 input vector F may bemultiplied by an 8×8 transform matrix D: F=DF. In the case of the 2-DDCT, the input vector F is replaced with an 8×8 matrix F, and the DCT isperformed as DFD″, where D′ is the transpose of D. Put

${D = {\sum\limits_{k = 0}^{n}{{}_{}^{}{}_{}^{}}}},$

where _(d)D_(k), the “detail transform”, is the kth refinement to anapproximation to D. Put

${D_{m} = {{\sum\limits_{k = 0}^{m}{{{}_{}^{}{}_{}^{}}{\mspace{11mu} \;}{and}\mspace{14mu} {\hat{F}}_{m}}} = {D_{m}F}}};$

i.e., the mth approximation to {circumflex over (F)}. Those skilled inthe art will recognize that the 8×8 matrix of input samples andcorresponding 8×8 transform matrix could be replaced with N₁×N₂ matrixof input samples using N₁×N₁ transform on the left and N₂×N₂ transformon the right.

However, because the scaled terms merely approximate the constants inthe transform equations, some error is accepted to keep the precisionconfined to a fixed number of bits or to minimize the number ofoperations. If the resulting numbers are further from a decisionboundary (e.g., a threshold value or a quantization boundary) than themaximum possible error, the result will not affected by theapproximations. Nevertheless, faster transforms may then be obtained byrefining the incremental calculations to obtain additional precision ifthe resulting numbers are determined during the incremental calculationsto require additional precision.

FIG. 3 illustrates a flow chart 300 for providing faster transformsusing corrective action to provide faster transform calculations anddecreased execution times. At least one number resulting from anincremental calculation using transform constants in a transform istested 310. Then, based upon the testing, when to perform a correctiveaction is determined 320. Once it is determined that a corrective actionis to be performed, the corrective action is performed 330.

A first example of refinement occurs when each _(d)D_(k) adds at leastone additional bit of precision to the transform performed by D. Asecond example occurs when at least one element of the transform vector,{circumflex over (F)}, is assumed to be very small, so that an entirerow of _(d)D_(k) may be approximated as zero, enabling us to skip thecalculation of that at least one element of F.

In the first example, it is often the case that all of the D_(k) axeinvertible; i.e., a matrix D_(k) ⁻¹ exists such that D_(k)D_(k) ⁻¹=D_(k)⁻¹D_(k)=I, the identity matrix, which has ones on the upper left tolower right diagonal, and zeros elsewhere. In this case, it may be notedthat

{circumflex over (F)} _(m+1) =D _(m+1) D _(m) ⁻¹ {circumflex over (F)}_(m)=(I+ _(d) D _(m+1) D _(m) ⁻¹){circumflex over (F)} _(m)

(where I is the identity matrix); i.e., the additional step of precisionis provided by performing one more step of transform to the transformedcoefficients. Using this additional step transform to add the precisionis the first embodiment of refinement provided by this invention, sinceit saves performing both IDCT and subsequent DCT: the matrix for(I+_(d)D_(m+1)D_(m) ⁻¹) can be calculated ahead of time as a matrixR_(m+1), so that {circumflex over (F)}_(m+1)=R_(m+1){circumflex over(F)}_(m), a single-step transform on F_(m).

The second example of refinement requires a different approach. Considera specific example, where the 2-D transform has already been performedwith high precision in the first dimension, FD′, _(d)D₀ has its 8th rowzero, and _(d)D₁=D−_(d)D₀. Then _(d)D₀ is not invertible; i.e., there isno way to recover the original 8 columns of FD′ from _(d)D₀FD′ (thisfollows from the fact that finding FD' from _(d)D₀FD′ may be viewed as 7equations in 8 unknowns). However, if an assumption is made for one ofthe 8 columns of FD″, then the other 7 columns can be estimated from_(d)D₀FD′, contingent on the assumption for the 8th column. A reasonableassumption is that the 8th column contains small elements that may beapproximated as zero, since the higher-numbered transformed values tendto be less significant in real images than the lower-numberedtransformed values. Then _(d)D₀ may be treated as an 8×7 matrix(ignoring the zero row), the pseudo-inverse, _(d){tilde over (D)}₀, (asis well-known in the literature) is found by

_(d) {tilde over (D)} _(o)(_(d) D′ _(0d) D ₀)⁻¹ _(d) D _(o)′

with an 8th row of zeros inserted for the assumed 8th coefficients. Thisgives an 8×8 approximation for _(d)D₀ ⁻¹ so that we can approximate

{circumflex over (F)} ₁=(I+ _(d) D _(1d) {tilde over (D)} ₀){circumflexover (F)} ₀

This approximate refinement is the second embodiment of the refinementinvention, which saves the cycles of the IDCT followed by the DCT, as inthe first example.

The abort procedure is used to determine when a calculation can beterminated before its completion to save cycles, when the result of thecalculation is projected to be too small, so that it wilt be quantizedto zero. One example of the application of the abort procedure appearsin example 2 above, where at least one low-magnitude transformcoefficient is not calculated, being essentially equivalent to settingthe corresponding row or rows of the transform matrix to zero. Anotherexample is stopping a calculation with limited precision, whenadditional transform precision is projected to provide negligibleadditional information in the transformed values; e.g., when the resultof the calculation is projected to be small. An alternative methodinvolves testing the magnitude of the sums and/or differences of some ofthe inputs to the transform. For example, for the FDCT, the followingequation calculates the second transform coefficient:

2S(2)=C ₂ d ₀₇₃₄ +C ₆ d ₁₆₂₅

where d₀₇₃₄=s₀₇−s₃₄ and d₁₆₂₅=s₁₆−s₂₅, notation from Pennebaker andMitchell's JPEG text. The magnitudes of these values can be tested forimpact on subsequent processing of the transform coefficients. In thisexample, if S(2) is less than the magnitude of Q/2 (where Q is aquantization value for S(2)), then S(2) will be quantized to zero. Thistranslates to a test of whether d₀₇₃₄ is less than Q/(2C₂) in magnitudeand d₁₆₂₅ is less than Q/(2C₆) in magnitude. If this test is met, thenthe calculation for S(2) can be aborted, and S(2) set to its quantizedvalue of zero. This method of testing sums and/or differences of theinput values can be extended to all of the equations for the FDCT.

It is not obvious how to turn a comparison such as −{circumflex over(T)}<{circumflex over (F)}<{circumflex over (T)} (a term-by-term rangecheck for the members of vector or matrix {circumflex over (F)}), wherethe elements of {circumflex over (T)} are all non-negative, into aterm-by-term comparison of the elements of F, −T<F<T, where the elementsof T are all non-negative, and where satisfying the test on F issufficient to satisfy the test on {circumflex over (F)}. The difficultyarises from the fact that the DCT employs both positive and negativeoperations, which destroys the term-by-term ordering in the equation.Specifically, it cannot be said that −{circumflex over (T)}<{circumflexover (F)}<{circumflex over (T)} implies that −D⁻¹{circumflex over(T)}<F<D⁻¹{circumflex over (T)}.

Thus, the abort involves terminating the precision of an operation whenadditional transform precision is projected to have an acceptable ornegligible effect on the results of the subsequent processingoperations, e.g., quantization or comparison. For example, thecoefficients of the DCT can be scaled by an integer and approximated assums of powers of 2. For the odd terms, one of these approximations isas follows:

${41\; D} = {{41\begin{bmatrix}C_{1} & C_{3} & C_{5} & C_{7} \\C_{3} & {- C_{7}} & {- C_{1}} & {- C_{5}} \\C_{5} & {- C_{1}} & C_{7} & C_{3} \\C_{7} & {- C_{5}} & C_{3} & {- C_{1}}\end{bmatrix}} \approx {{32\begin{bmatrix}1 & 1 & 1 & 0 \\1 & 0 & {- 1} & {- 1} \\1 & {- 1} & 0 & 1 \\0 & {- 1} & 1 & {- 1}\end{bmatrix}} + {8\begin{bmatrix}1 & 0 & {- 1} & 1 \\0 & {- 1} & {- 1} & 1 \\{- 1} & {- 1} & 1 & 0 \\1 & 1 & 0 & {- 1}\end{bmatrix}} + {2\begin{bmatrix}0 & 1 & {- 1} & 0 \\1 & 0 & 0 & 1 \\{- 1} & 0 & 0 & 1 \\0 & 1 & 1 & 0\end{bmatrix}} + \begin{bmatrix}0 & 0 & {- 1} & 0 \\0 & 0 & 0 & 1 \\{- 1} & 0 & 0 & 0 \\0 & 1 & 0 & 0\end{bmatrix}}}$

which we write as (using the notation from above),

41D≈2⁵ _(d) D ₀+2³ _(d) D ₁+2¹ _(d) D ₂+2⁰ _(d) D ₃

Also, as mentioned above, all of the matrices above, and theirsequential sums, are invertible. Now, if |j{circumflex over (F)}|<<1,i.e., the jth element of {circumflex over (F)} is very small inmagnitude, then j (32_(d)D₀F) should be small. If it is not, then j((8_(d)D₁+2_(d)D₂+_(d)D₃) F) will not be able to cancel it out to makethe final result small. The relative magnitudes of the results of thecalculations may be checked. If one of the transform values, for one ofthe intermediate precisions, is small compared to the other values, oris small compared to some pre-determined threshold, then subsequentrefinements for that transform value can be aborted.

FIG. 4 illustrates a flow chart 400 of the abort method according to thepresent invention that demonstrates aborting further iterations of thetransform coefficient calculation process. In FIG. 4, at least oneincrementally calculated number is tested 410. If certain criteria aremet, further calculations are aborted 420. The incremental calculationof a transform coefficient may be aborted when an error resulting fromterminating the incremental calculation is acceptable. For example, theincremental calculation may be terminated when a determination is madethat the incremental calculation will result in a number that isprojected to be within a predetermined range, e.g., a transformcoefficient that does satisfy a precision requirement. Alternatively,the incremental calculation of the transform coefficient may be abortedwhen a transform coefficient is going to be within a predetermined rangeof zero.

FIG. 5 is a flow chart 500 of the testing of the at least oneincrementally calculated number. In FIG. 5, the incremental calculationis tested to determine when a refinement to the transform coefficientwill not change the result 510. This testing may be carried out in atleast two ways as shown in FIG. 5. A refinement to the transformcoefficient may be determined not to change the result when, afterchecking the relative magnitudes of the results of the incrementalcalculations, an intermediate calculation of at least one transformcoefficient is small compared to the intermediate calculation of anothertransform coefficient 520. Alternatively, a refinement to the transformcoefficient may be determined not to change the result when, afterchecking the magnitude of the results of at least one incrementalcalculation, at least one intermediate calculation of the transformcoefficient is less than a predetermined threshold 530.

FIG. 6 is a flow chart 600 of the refinement method according to thepresent invention. First, a determination is made whether the transformrequires more precision 610. The transform is a transform matrix,wherein a refinement matrix may be used to increase precision of theincremental calculation of the transform coefficients. When moreprecision is required, a refinement matrix is applied to the transform620. The refinement matrix is generated offline or at initialization andis based on approximately calculated transform constants.

FIG. 7 illustrates a flow chart 700 of a first method for generating arefinement matrix. First, it is recognized that an approximate transformis invertible 710. The refinement matrix given by I+_(d)D_(m+1)D_(m) ⁻¹is generated 720. Then, the transform is structured for efficientcomputation 730.

However, as described above, when _(d)D₀ is not invertible; there is noway to recover the original 8 columns of FD' from _(d)D₀FD″. FIG. 8 is aflow chart 800 showing a second method for generating a refinementmatrix when _(d)D₀ is not invertible. It is first recognized thatrecovery of the nth column of a transform matrix for generating thetransform is impossible 810. A pseudo inverse for a portion of thetransform matrix is calculated 820. Then, an approximation for therefinement matrix is generated using the pseudo inverse for thetransform matrix 830. The approximation of the refinement matrixcomprises I+_(d)D_(1d){tilde over (D)}₀.

FIG. 9 illustrates a block diagram 900 of a printer 920 according to thepresent invention. In FIG. 9, the printer 920 receives image data 912from a host processor 910. The image data 912 is provided into memory930 where the image data may be arranged into N₁×N₂ block samples. TheN₁×N₂ block samples are then processed by a processor 940, such as araster image processor. The raster image processor 940 provides acompressed print stream representing the image data to a printheaddriving circuit 950. The printhead driving circuit 950 then controls theprinthead 960 to generate a printout 970 of the image data.

The process illustrated with reference to FIGS. 1-3 may be tangiblyembodied in a computer-readable medium or carrier 990, e.g. one or moreof the fixed and/or removable data storage devices illustrated in FIG.9, or other data storage or data communications devices. The computerprogram may be loaded into the memory 992 to configure the processor 940of FIG. 9, for execution. The computer program comprises instructionswhich, when read and executed by the processor 940 of FIG. 9, causes theprocessor 940 to perform the steps necessary to execute the steps orelements of the present invention.

FIG. 10 illustrates a data analyzing system 1000 according to thepresent invention. In FIG. 10, a transformer 1010 receives a block ofdata 1012 to be analyzed. The transformer 1010 uses transform equations1020 to generate transformed data 1024. Transform equations 1020 aresplit into at least one sub-transform having at least two transformconstants. The at least two transform constants for each collection arescaled independently of the other collections with a scaling term tomaintain a substantially uniform ratio between the at least twotransform constants within the at least one collection, wherein thescaling term is chosen according to a predetermined cost function. Thetransformed data 1024 may then be quantized by an optional quantizer1030.

FIG. 11 illustrates another data analyzing system 1100 according to thepresent invention. In FIG. 11, a transformer 1110 receives a block ofdata 1112 to be analyzed. The transformer 1110 uses transform equations1120 to generate transformed data 1124. Transform equations 1120 aresplit into at least one sub-transform having at least two transformconstants. The at least two transform constants for each collection arescaled independently of the other collections with a scaling term tomaintain a substantially uniform ratio between the at least twotransform constants within the at least one collection, wherein thescaling term may be chosen according to a predetermined cost function.The transformed data 1124 may then be compared to scaled comparisonvalues in an optional comparator 1130.

The foregoing description of the exemplary embodiment of the inventionhas been presented for the purposes of illustration and description. Itis not intended to be exhaustive or to limit the invention to theprecise form disclosed. Many modifications and variations are possiblein light of the above teaching. It is intended that the scope of theinvention be limited not with this detailed description, but rather bythe claims appended hereto.

1-16. (canceled)
 17. A printer, comprising: memory for storing imagedata; a processor for processing the image data to provide a printstream output; and a printhead driving circuit for controlling aprinthead to generate a printout of the image data; wherein theprocessor reduces errors of the transform by testing at least one numberresulting from an incremental calculation of transform coefficientsduring a transform, detecting whether the incremental calculation of thetransform coefficients will result in transform coefficients withunacceptable precision of the image data, and if determined to beneeded, refining the at least one number to obtain additional precisionof the image data.
 18. (canceled)
 19. The printer of claim 17 whereinthe transform comprises a transform matrix and wherein the transformerrefines the at least one number by applying a refinement matrix forincreasing precision of the incremental calculation of the transformconstants.
 20. The printer of claim 19 wherein the refinement matrixcomprises I+_(d)D_(m+1)D_(m) ⁻¹.
 21. The printer of claim 19 wherein therefinement matrix is based on approximately calculated transformconstants.
 22. The printer of claim 21 wherein the refinement matrix isgenerated offline or at initialization.
 23. The printer of claim 20wherein the refinement matrix is generated by recognizing that anapproximate transform is invertible, generating the refinement matrixgiven by I+_(d)D_(m+1)D_(m) ⁻¹, and structuring the transform forefficient computation.
 24. The printer of claim 20 wherein therefinement matrix is generated by recognizing that recovery of the nthcolumn of a transform matrix for generating the transform is impossible,calculating a pseudo inverse for a portion of the transform matrix andgenerating an approximation for the refinement matrix using the pseudoinverse for the transform matrix.
 25. The printer of claim 24 whereinthe approximation of the refinement matrix comprises I+_(d)D_(1d){tildeover (D)}₀.
 26. The printer of claim 17 wherein the transformer:determines whether an error resulting from terminating the incrementalcalculation is acceptable, and aborts the incremental calculation of atransform coefficient.
 27. The printer of claim 26 wherein thetransformer terminates the incremental calculation when a determinationis made that the incremental calculation will result in a number that isprojected to be within a predetermined range.
 28. The printer of claim27 wherein the number that is projected to be within a predeterminedrange comprises a transform coefficient that does satisfy a precisionrequirement.
 29. Claim 27 wherein the transformer terminates theincremental calculation when a refinement to the transform coefficientis determined not to change the result.
 30. The printer of claim 29wherein the transformer determines that a refinement to the transformcoefficient will not change the result when, after checking the relativemagnitudes of the results of the incremental calculations, anintermediate calculation of at least one transform coefficient is smallcompared to the intermediate calculation of another transformcoefficient.
 31. The printer of claim 29 wherein the transformerdetermines that a refinement to the transform coefficient will notchange the result when, after checking the magnitude of the results ofat least one incremental calculation, at least one intermediatecalculation of the transform coefficient is less than a predeterminedthreshold.
 32. The printer of claim 17 wherein the transformer:determines that a transform coefficient is going to be within apredetermined range of zero, and aborts the incremental calculation ofthe transform coefficient.
 33. An article of manufacture comprising aprogram storage medium readable by a computer, the medium tangiblyembodying one or more programs of instructions executable by thecomputer to perform a method for reducing errors during data processing,the method comprising: inputting data; testing at least one numberresulting from an incremental calculation of transform coefficientsduring a transform of the data; detecting whether the incrementalcalculation of the transform coefficients will result in transformcoefficients with unacceptable precision; and if determined to beneeded, refining the at least one number to obtain additional precision.34. (canceled)
 35. The article of manufacture of claim 33 wherein thetransform comprises a transform matrix and wherein the refiningcomprises applying a refinement matrix for increasing precision of theincremental calculation of the transform constants.
 36. The article ofmanufacture of claim 35 wherein the refinement matrix comprisesI+_(d)D_(m+1)D_(m) ⁻¹.
 37. The article of manufacture of claim 33further comprising generating at least one refinement matrix based onapproximately calculated transform constants.
 38. The article ofmanufacture of claim 37 wherein the generating at least one refinementmatrix is performed offline or at initialization.
 39. The article ofmanufacture of claim 37 wherein the generating the at least onerefinement matrix comprises recognizing that an approximate transform isinvertible, generating the refinement matrix given by I+_(d)D_(m+1)D_(m)⁻¹, and structuring the transform for efficient computation.
 40. Thearticle of manufacture of claim 37 wherein the generating the at leastone refinement matrix comprises: recognizing that recovery of the nthcolumn of a transform matrix for generating the transform is impossible;calculating a pseudo inverse for a portion of the transform matrix; andgenerating an approximation for the at least one refinement matrix usingthe pseudo inverse for the transform matrix.
 41. The article ofmanufacture of claim 40 wherein the approximation of the refinementmatrix comprises I+_(d)D_(1d){tilde over (D)}₀.
 42. The article ofmanufacture of claim 33 further comprising determining whether an errorresulting from terminating the incremental calculation is acceptable andaborting the incremental calculation of a transform coefficient.
 43. Thearticle of manufacture of claim 42 wherein the incremental calculationis terminated when a determination is made that the incrementalcalculation will result in a number that is projected to be within apredetermined range.
 44. The article of manufacture of claim 43 whereinthe number that is projected to be within a predetermined rangecomprises a transform coefficient that does satisfy a precisionrequirement.
 45. The article of manufacture of claim 43 wherein theincremental calculation is terminated when a refinement to the transformcoefficient is determined not to change the result.
 46. The article ofmanufacture of claim 45 wherein a refinement to the transformcoefficient is determined not to change the result when, after checkingthe relative magnitudes of the results of the incremental calculations,an intermediate calculation of at least one transform coefficient issmall compared to the intermediate calculation of another transformcoefficient.
 47. The article of manufacture of claim 45 wherein arefinement to the transform coefficient is determined not to change theresult when, after checking the magnitude of the results of at least oneincremental calculation, at least one intermediate calculation of thetransform coefficient is less than a predetermined threshold.
 48. Thearticle of manufacture of claim 33 further comprising determining that atransform coefficient is going to be within a predetermined range ofzero and aborting the incremental calculation of the transformcoefficient.
 49. A data analysis system, comprising: a data input;transform equations formed by testing at least one number resulting froman incremental calculation of transform coefficients during a transform,detecting whether the incremental calculation of the transformcoefficients will result in transform coefficients with unacceptableprecision, and if determined to be needed, refining the at least onenumber to obtain additional precision; a transformer for applying thetransform equations to perform a linear transform to decorrelate thedata into transform coefficients; and a data output.
 50. (canceled) 51.The data analysis system of claim 49 wherein the transform comprises atransform matrix and wherein the transformer refines the at least onenumber by applying a refinement matrix for increasing precision of theincremental calculation of the transform constants.
 52. The dataanalysis system of claim 51 wherein the refinement matrix comprisesI+_(d)D_(m+1)D_(m) ⁻¹.
 53. The data analysis system of claim 51 whereinthe refinement matrix is based on approximately calculated transformconstants.
 54. The data analysis system of claim 53 wherein therefinement matrix is generated offline or at initialization.
 55. Thedata analysis system of claim 52 wherein the refinement matrix isgenerated by recognizing that an approximate transform is invertible,generating the refinement matrix given by I+_(d)D_(m+1)D_(m) ⁻¹, andstructuring the transform for efficient computation.
 56. The dataanalysis system of claim 52 wherein the refinement matrix is generatedby recognizing that recovery of the nth column of a transform matrix forgenerating the transform is impossible, calculating a pseudo inverse fora portion of the transform matrix and generating an approximation forthe refinement matrix using the pseudo inverse for the transform matrix.57. The data analysis system of claim 56 wherein the approximation ofthe refinement matrix comprises I+_(d)D_(1d){tilde over (D)}₀.
 58. Thedata analysis system of claim 49 wherein the transformer determineswhether an error resulting from terminating the incremental calculationis acceptable and aborts the incremental calculation of a transformcoefficient.
 59. The data analysis system of claim 58 wherein thetransformer terminates the incremental calculation when a determinationis made that the incremental calculation will result in a number that isprojected to be within a predetermined range.
 60. The data analysissystem of claim 59 wherein the number that is projected to be within apredetermined range comprises a transform coefficient that does satisfya precision requirement.
 61. The data analysis system of claim 59wherein the transformer terminates the incremental calculation when arefinement to the transform coefficient is determined not to change theresult.
 62. The data analysis system of claim 61 wherein the transformerdetermines that a refinement to the transform coefficient will notchange the result when, after checking the relative magnitudes of theresults of the incremental calculations, an intermediate calculation ofat least one transform coefficient is small compared to the intermediatecalculation of another transform coefficient.
 63. The data analysissystem of claim 61 wherein the transformer determines that a refinementto the transform coefficient will not change the result when, afterchecking the magnitude of the results of at least one incrementalcalculation, at least one intermediate calculation of the transformcoefficient is less than a predetermined threshold.
 64. The dataanalysis system of claim 49 wherein the transformer determines that atransform coefficient is going to be within a predetermined range ofzero and aborts the incremental calculation of the transformcoefficient.